The Singularly Perturbed Turning Point Problem: A Spectral Approach.
Abstract
Consider the two-point boundary value problem -Epsilon u'' + xu prime - lambda u = 0, u(-1) = A, u(1) = B, with 0 < Epsilon <<1. For certain values of Lambda, namely the eigenvalues, the problem need not have a solution; for Epsilon --> +0 those eigenvalues tend to the non-negative integers. Although for values near such an eigenvalue the solution exists, it is very sensitive to small changes in lambda, and for a long time it has been unknown how to construct a rigorous asymptotic approximation to it. Yet an asymptotic approximation is of interest since the equation and its generalizations offer a deterministic model for the motion of a particle in a potential field, which executes a random walk under influence of small random forces. In this paper we construct asymptotic approximations to solutions of such problems, in which the coefficient of u prime has one or several zeros (turning points) and we prove their validity.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1980
- Accession Number
- ADA096657
Entities
People
- Pieter De Groen
Organizations
- University of Wisconsin–Madison